Interlocked open linkages with few joints
Proceedings of the eighteenth annual symposium on Computational geometry
Interlocked open and closed linkages with few joints
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
Acute Triangulations of Polygons
Discrete & Computational Geometry
Locked and unlocked chains of planar shapes
Proceedings of the twenty-second annual symposium on Computational geometry
On straightening low-diameter unit trees
GD'05 Proceedings of the 13th international conference on Graph Drawing
On unfolding lattice polygons/trees and diameter-4 trees
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Folding equilateral plane graphs
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
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Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a six-edge tree can interlock with a four-edge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length).